Chapter 54 - David Hilbert, report on algebraic number fields (‘Zahlbericht’) (1897)

Abstract

In his report on algebraic number fields called Zahlbericht, Hilbert summed up the current state of knowledge in algebraic number theory, at the same time enriching and organizing the subject in ways that were to influence developments for decades. The first part of Zahlbericht discusses the basic arithmetic theory of a general finite extension of the field of rational numbers: integers, ideals, discriminant, units, and ideal classes. The second part deals with the decomposition of primes in a Galois extension: decomposition group and inertia group, and the corresponding subfields. The third Part of the Zahlbericht deals with quadratic fields. The theory of cyclotomic fields follows suit in the fourth Part, including the theory of circular units, and together with Hilbert's proof of the Kronecker–Weber Theorem to the effect that every abelian extension of the rational numbers is contained in a suitable cyclotomic field. Then follows a discussion of normal bases and what Hilbert calls their “associated root numbers,” that is, generalized Gaussian sums. The major reason reason for its great impact, apart from its striking expositional quality, was the fact that Hilbert was able to present current (algebraic) number theory as a leading mathematical discipline in tune with what he saw as the dominating values of the time.